Linearly Dependent Functions

The n functions f_1(x), f_2(x), ..., f_n(x) are linearly dependent if, for some c_1, c_2, ..., c_n in R not all zero,

for all x in some interval I. If the functions are not linearly dependent, they are said to be linearly independent. Now, if the functions and in C^(n-1) (the space of functions with n-1 continuous derivatives), we can differentiate (1) up to n-1 times. Therefore, linear dependence also requires

where the sums are over i=1, ..., n. These equations have a nontrivial solution iff the determinant

where the determinant is conventionally called the Wronskian and is denoted W(f_1,f_2,...,f_n).

If the Wronskian !=0 for any value c in the interval I, then the only solution possible for (2) is c_i=0 (i=1, ..., n), and the functions are linearly independent. If, on the other hand, W=0 over some range, then the functions are linearly dependent somewhere in the range. This is equivalent to stating that if the vectors V[f_1(c)], ..., V[f_n(c)] defined by

are linearly independent for at least one c in I, then the functions f_i are linearly independent in I.

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References

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Cite this as:

Weisstein, Eric W. "Linearly Dependent Functions." From MathWorld--A Wolfram Resource. https://mathworld.wolfram.com/LinearlyDependentFunctions.html

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